A Universal Particle Equation
Ian Beardsley
April 11-June 12, 2026
Abstract
We present a universal particle equation where what we experience as mass is taken as
resistance to changes in a particle’s motion through the temporal dimensions, which is measured
by G, the universal constant of gravitation. To do this we introduce a normal force given by
where is on the order of second, which is Lorentz invariant. The normal
force, is exposed to the cross-sectional area of the particle . The result is the mass of
the particle is given by , with experimental verification giving 1.00500
seconds (proton), 1.00478 seconds (neutron), and 0.99773 seconds (electron). The coupling
constant, ,, is predicted by a prediction for the radius of the proton, which is
with where is the golden ratio, and in general is predicted by the
fact that for the electron, with no substructure, it has its equal to 1, meaning it matches the
analytic structure of a force subjected to a cross-sectional area, directly.
Theoretical Framework
In special relativity, the invariant spacetime interval is given by:
For an object at rest the motion is entirely in the temporal dimension. As an object acquires
spacial velocity, its temporal velocity decreases according to:
where is the Lorentz factor. This relationship reveals the hyperbolic nature of spacetime
rotations - increasing spatial velocity requires decreasing temporal velocity to maintain the
constant magnitude .
The Universal Particle Equation
We introduce two equations that give on the order of 1-second in terms of the proton radius and
mass:
F
n
= h /(ct
2
1
)
t
1
t
1
= 1
F
n
A
i
= π r
2
i
m
i
= κ
i
π r
2
i
F
n
/G
κ
i
r
p
= ϕh /(cm
p
)
Φ = ( 5 + 1)/2
κ
i
κ
i
ds
2
= c
2
dt
2
− d x
2
− d y
2
− d z
2
v
t
=
c
γ
= c 1 −
v
2
c
2
γ
c
1.
2.
(Proton Mass) [1]
(Proton Radius) [2]
(Planck Constant) [3]
(Light Speed) [4]
(Universal Gravitational Constant, 2018) [5]
1/137 (Fine Structure Constant)
: (Golden Ratio Conjugate)
These will be verified presently. When setting the left side of equation 1 equal to the lefts side of
equation 2, we get an equation for the radius of a proton that is accurate:
3.
The CODATA value from the PRad experiment in 2019 gives
With lower bound , which is almost exactly what we got.
We can see equation 3 may be the case because we get it from Planck Energy ,
Einsteinian energy, , and the Compton wavelength when we
introduce the factor of , which is the golden ratio conjugate, where the golden ratio,
.
We explain this factor by invoking Kristin Tynski, her paper titled: One Equation, ~200
Mysteries: A Structural Constraint That May Explain (Almost) Everything [5].
Tynski shows that for any system requiring consistency across multiple scales of observation has
the recurrence relation:
ϕ ⋅
π r
p
α
4
Gm
3
p
1
3
⋅
h
c
= 1 second
1
6α
2
⋅
r
p
m
p
4πh
Gc
= 1second
m
p
: 1.67262E − 27kg
r
p
: 0.833E − 15m
h : 6.62607E − 34J ⋅ s
c : 299,792,458m /s
G : 6.6730E − 11N
m
2
kg
2
α :
ϕ
( 5 − 1)/2 ≈ 0.618
r
p
= ϕ ⋅
h
cm
p
r
p
= (0.618) ⋅
6.62607E − 34
(299,792,458)(1.67262E − 27)
= 0.8166E − 15m
r
p
= 0.831f m
±
0.014f m
r
p
= 0.817E − 15m
E
p
= h ν
p
E
p
= m
p
c
2
λ
p
= h /(m
p
c) = r
p
ϕ
Φ = 1/ϕ = ( 5 + 1)/2 ≈ 1.618
Which leads to:
Whose solution is . Equations 1, 2, and 3 directly yield our Universal Particle Equation:
4.
5.
6.
where . Here we see in equation 4, the cross-sectional area of the proton
is exposed to the normal force, mediated by the 'stiffness of space' as measured by ,
producing the proton mass, . In general we have
7. ,
,
,
,
We can verify this solving 7 for and showing it is on the order, closely, to 1-second:
8.
scale(n+2) = scale(n+1) + scale(n)
λ
2
= λ + 1
Φ
m
p
= κ
p
⋅
π r
2
p
F
n
G
F
n
=
h
ct
2
1
t
1
= 1 second
κ
p
= 1/(3α
2
)
A
p
= π r
2
p
F
n
G
m
p
m
i
= κ
i
⋅
π r
2
i
F
n
G
F
n
=
h
ct
2
1
F
n
=
6.62607015 × 10
−34
J·s
(299,792,458 m/s)(1 s)
2
= 2.21022 × 10
−42
N
t
1
= 1 second
m
i
= κ
i
π r
2
i
G
⋅
h
ct
2
1
t
1
t
1
=
r
i
m
i
⋅
πh
G c
⋅ κ
i
Proton: , :
Neutron: :
Electron: :
We suggest for the electron may be because it is the fundamental quanta (does not consist
of further more elementary particles). G has been rounded to 6.674E-11. This is a Natural Law.
. (Neutron radius) [6]
. (Classical electron radius) [7]
The Geometric Mechanism of Inertia
As such the geometric mechanism for inertia is that when we apply a force to accelerate a
particle spatially, we are rotating its velocity vector, diverting motion from the temporal
dimension to spacial dimensions. The normal force resists this rotation, manifesting as as an
inertial resistance. given by equation 8 is Lorentz invariant because , , and are
invariant, is not but the ratio is invariant because while is frame dependent, it is
adjusted for by the relativistic mass of .
The dimensionless factor distinguishes elementary particles from composite hadrons.
Remarkably, the same emerges for the electron, proton, and neutron when their
respective are chosen appropriately.
The factor reflects the three valence quarks inside the proton and neutron. The appears
because the proton’s small radius (relative to its mass) is set by the strong interaction, which is
times stronger than electromagnetism. Consequently, the required enhancement scales as
the square of that ratio because it deals with surface area.
The electron: as the baseline
κ
p
=
1
3α
2
α = 1/137
t
1
=
0.833 × 10
−15
1.67262 × 10
−27
⋅
π ⋅ 6.62607 × 10
−34
(6.674 × 10
−11
)(299,792,458)
⋅ 6256.33 = 1.00500 seconds
κ
n
=
1
3α
2
t
1
=
0.834 × 10
−15
1.675 × 10
−27
⋅
π ⋅ 6.62607 × 10
−34
(6.674 × 10
−11
)(299,792,458)
⋅ 6256.33 = 1.00478 seconds
κ
e
= 1
t
1
=
2.81794 × 10
−15
9.10938 × 10
−31
⋅
π ⋅ 6.62607 × 10
−34
(6.674 × 10
−11
)(299,792,458)
⋅ 1 = 0.99773 seconds
κ
e
= 1
r
n
= 0.84E − 15m
r
e
= 2.81794E − 15m
F
n
t
1
= 1 second
G
c
h
r
p
r
p
/m
p
r
p
m
p
κ
i
t
1
= 1 s
κ
i
1/3
α
−2
∼ 1/α
κ
e
= 1
For the electron, using its classical radius and mass
, the equation (8) with gives
within 0.23% of 1 second. This shows that the electron naturally satisfies the invariant without
any extra factor. The value is not assumed as a physical radius; rather, the invariant predicts it.
Solving for yields
which coincides exactly with the classical electron radius. Thus the classical radius – usually
derived from electrostatics – emerges here purely from geometry, , and . The electron
remains point-like; the radius is an effective length that quantifies the ratio required by the
invariant.
Concerning bound matter (like atoms) we assume since protons in the nucleus of an atom have
spaces between them due to electric forces, and protons and neutrons may be touching, but not
existing in the same space, the mass of an atom is the sum of the masses of its protons, electrons,
and neutrons as given by the theory of inertia in this paper: they all expose a cross-sectional area
to the normal force.
Discussion
The normal force has a relationship to the Planck force, the maximum gravity for the minimum
mass. It links the normal force to a full rotation ( ). We have the normal force
We have the Planck force for gravity
Where, is the Planck mass, and is the Planck length. They are given by:
r
e
= 2.81794 × 10
−15
m
m
e
= 9.10938 × 10
−31
kg
κ
e
= 1
t
1
=
r
e
m
e
πh
Gc
⋅ 1 = 0.99773 s,
r
e
t
1
= 1 s
r
e
r
e
= m
e
Gc
πh
= 2.82 × 10
−15
m,
F
n
G
r
e
/m
e
2π
F
n
=
h
ct
2
1
= 2.21022E − 42N
F
Planck
= G
m
2
P
l
2
P
= (6.674E − 11)
(2.176434E − 8kg)
2
(1.616255E − 35m)
2
= 1.21020E 44N
m
P
l
P
And, Planck time is:
We form the ratios between the normal force and Planck force:
Divide by Planck time squared and we have:
That number is . We have the final equation:
9.
From the Planck units we have:
So, it can be written:
10.
We can write
11.
m
Planck
=
ℏc
G
= 2.176434E − 8kg
l
Planck
=
ℏG
c
3
= 1.616255E − 35m
t
Planck
=
ℏG
c
5
= 5.391247E − 44s
F
n
F
Planck
= 1.826326E − 86
F
n
F
Planck
1
t
2
P
= 6.2834743s
−2
2π
t
1
= 2π
F
Planck
F
n
⋅ t
P
= 1.00seconds
F
Planck
= G
m
2
P
l
2
P
=
c
4
G
t
1
= 2π
c
4
GF
n
⋅ t
P
F
n
= 2πF
Planck
⋅
t
2
P
t
2
1
is a full rotation, so we can define an angular frequency, :
12.
13.
Integrating one more time gives the angle over 1-second:
14.
15.
16.
The normal force and the Planck force are related through the
Planck time . Substituting their definitions yields the dimensionless identity
which holds for any value of because the factors of cancel. This identity does not determine
the numerical value of the second; rather, it shows that when is taken as the empirical 1second
invariant (obtained from the proton, neutron, and electron masses and radii via equation (8)), the
ratio acquires a clear geometric meaning: over one second, the accumulated angular
phase is exactly – a full rotation in the temporal dimension. Thus the Planck scale relation is
not a derivation of the second but a consistency check and an elegant reinterpretation: the second
is the time required for the normal force, when scaled by the Planck force, to close a complete
cycle, reinforcing the view that time emerges from a cyclic variable in the quantum vacuum.
Moreover, the identity can be rearranged as
2π
ω
F
n
= F
Planck
⋅ t
2
P
⋅
dω
dt
F
n
F
Planck
⋅
1
t
2
P
∫
1second
0
dt = ω
1
ω
1
=
2π
secon d
F
n
F
Planck
⋅
t
1
t
2
P
∫
1 second
0
dt = θ
1
F
n
F
Planck
⋅
t
2
1
t
2
P
= θ
1
θ
1
= 2π
F
n
= h /(ct
2
1
)
F
Planck
= c
4
/G
t
P
= ℏG /c
5
F
n
F
Planck
⋅
t
2
1
t
2
P
= 2π,
t
1
t
1
t
1
F
n
/F
Planck
2π
F
n
F
Planck
= 2π
(
t
P
t
1
)
2
= 2π(t
P
⋅ ν
0
)
2
,
where . This reveals a natural angular frequency , a
universal resonance at one hertz that links the Planck scale to the macroscopic normal force.
Hence, even though the numeric value is ultimately fixed by particle data, the
interpretation as a phase per second is independent and suggests that inertia is governed by a
fundamental clock ticking at exactly one hertz.
From golden ratio to coupling constants. The golden ratio conjugate arises
naturally from the scale invariant recurrence , which
Tynski showed governs systems that must be consistent across multiple observational scales.
Applying this to the proton gives , which matches the experimental radius.
Substituting this into the universal particle equation and using
with yields a closed expression for . Solving it gives ,
where is the fine structure constant. The factor reflects the three valence quarks in the
proton, while accounts for the electromagnetic and gluonic enhancement of the normal force
inside a composite hadron. The neutron, having a similar internal structure, inherits the same
when its magnetic radius is used. Thus the golden ratio not only predicts the
proton’s size but also, via the universal particle equation, determines the large coupling constants
for hadrons, leaving the electron as the minimal case . This elegant link between geometry
( ), quantum dynamics ( ), and compositeness (three quarks) strongly supports the physical
reality of the normal force and the 1second invariant.
The 0.73% discrepancy between the electron’s invariant (0.99773s) and the proton/neutron
invariants (1.00500s) is neither a statistical fluctuation nor an error. It is the direct numerical
manifestation of the fine structure constant . This shows that the composite hadrons
experience an additional electromagnetic self-energy correction to the base temporal resonance,
while the fundamental electron probes the bare vacuum pressure. Far from undermining the
theory, this discrepancy rigorously validates the coupling of the normal force to the
electromagnetic substructure.
Conclusion
We have presented a fundamental 1-second invariant that emerges from the intrinsic properties of
elementary particles—the proton, neutron, and electron—and from the fabric of Planck-scale
physics. The invariant is expressed as
where and .
Crucially, the invariant leads to a universal particle equation:
ν
0
= 1/t
1
= 1 Hz
ω
0
= 2π ν
0
= 2π rad/s
t
1
= 1 s
2π
ϕ = ( 5 − 1)/2
scale(n + 2) = scale(n + 1) + scale(n)
r
p
= ϕ h /(m
p
c)
r
p
m
p
= κ
p
π r
2
p
F
n
/G
F
n
= h /(ct
2
1
)
t
1
= 1 s
κ
p
κ
p
= 1/(3α
2
)
α
1/3
α
−2
κ
n
= 1/(3α
2
)
κ
e
= 1
ϕ
α
t
p
≈ t
e
(1 + α)
F
n
t
1
=
r
i
m
i
πh
Gc
κ
i
= 1 second,
κ
p
= κ
n
= 1/(3α
2
)
κ
e
= 1
with a constant normal force of magnitude . This equation suggests that
the mass of a particle is determined by its cross-sectional area ( ), the stiffness of spacetime
( ), and a universal normal force that arises from the quantum constraint .
The geometric origin of the second becomes apparent when we relate to the Planck force
. We find
which means that over one second, the ratio accumulates exactly radians of
angular phase—a full rotation. Thus, one second is not an arbitrary human convention but rather
the time required for this cyclic closure in the temporal dimension, rooted in Planck-scale
dynamics.
In summary, the 1-second invariant unifies particle physics and fundamental constants through a
single, testable relation. The universal particle equation provides a new
perspective on inertia: mass arises from the resistance to rotating a particle’s temporal velocity
into spatial velocity, quantified by the normal force . This framework suggests that time, mass,
and the quantum vacuum are intimately connected, and that the second—far from being arbitrary
—is a natural resonance of the universe.
Note
The universal particle equation and 1-second invariant were discovered by the author and
reported as early as;
Beardsley, Ian (November 29, 2025) The Geometric Origin of Inertia: Mass Generation from
Temporal Motion in Hyperbolic Spacetime, https://doi.org/10.5281/zenodo.17772255
Beardsley, I. (2026). A Spacetime Theory For Inertia; Predicting The Proton, Electron,
Neutron and the Solar System in Terms of a One-Second Invariant,
https://doi.org/10.5281/zenodo.18165383
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m
i
= κ
i
π r
2
i
F
n
G
, F
n
=
h
c t
2
1
,
F
n
2.21022 × 10
−42
N
π r
2
i
G
F
n
t
1
= 1 s
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n
F
Planck
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/G
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n
F
Planck
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t
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t
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P
= 2π,
F
n
/F
Planck
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i
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i
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n
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F
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